The Hunger Games takes place in North America, where the United States used to be; the new nation is named Panem. A terrible war has decimated the country, perhaps even the world, and so there are only a few small enclaves of humans left on the continent. Although the specific mountain that District 12 mines is not named, the region is shown in the text:

District 12 was in a region known as Appalachia. Even hundreds of years ago, they mined coal here.(Collins, The Hunger Games, Google Books)

This places District 12 in the Eastern United States, probably on the West side of the Appalachian Mountain range. The fan website linked below contains several pieces of information about how Panem is laid out, as well as with a few maps; in addition, Suzanne Collins reads Katniss's voice with a Southern accent, meaning that District 12 is probably near the Southern end of the Appalachian Mountain range.

The Hunger Games take place in Panem and the districts are said to be in North America. If you look at a map there is a chance you could figure out where it is, but they never gave a specific answer as to where it is or what state what districts are since this is probably many many years into the future probably even 5 centuries into the future.